Generate and return the natural logarithm of the volume of an \(\ndim\)dimensional ellipsoid.
More...
Generate and return the natural logarithm of the volume of an \(\ndim\)dimensional ellipsoid.
See the documentation of pm_ellipsoid for computational and algorithmic details.
 Parameters

[in]  gramian  : The input matrix of the same type and kind as the output logVolEll , containing the upper triangle and diagonal of the representative Gramian matrix of the ellipsoid.

 Returns
logVolEll
: The output scalar of,

type
real
of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
containing natural logarithm of the volume of the \(\ndim\)dimensional hyperellipsoid.
Possible calling interfaces ⛓
Generate and return the natural logarithm of the volume of an dimensional ellipsoid.
This module contains classes and procedures for setting up and computing the properties of the hyper...
 Warning
 If the Cholesky factorization of the input Gramian fails, the procedures of this generic interface will abort the program by calling
error stop
.

The condition
size(gramian, 1) == size(gramian, 2)
must hold for the corresponding input arguments.
The condition \(0. < \left\ms{gramian}\right\) must hold for the corresponding input arguments.
In other words, the input Gramian must be a positive definite matrix.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1
.

The
pure
procedure(s) documented herein become impure
when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1
.
By default, these procedures are pure
in release
build and impure
in debug
and testing
builds.
 Note
 Computing the volume of an ellipsoid using its Gramian in a fixed dimension as implemented by the procedure of this generic interface are computationally costly.
The unnecessary costs can be eliminated by precomputing the natural logarithm of the volume of the unit ball in the desired dimension once via setLogVolUnitBall and adding to it the sum of the natural logarithms of the diagonal elements of the Cholesky factorization of the representative Gramian matrix of the ellipsoid.
 See also
 getLogVolEll
Example usage ⛓
9 real,
allocatable :: gramian(:,:)
11 type(display_type) :: disp
15 call disp%show(
"gramian = reshape([1., 0., 0., 1.], [2,2]) ! correlation 0.")
16 gramian
= reshape([
1.,
0.,
0.,
1.], [
2,
2])
17 call disp%show(
"exp(getLogVolEll(gramian)) ! correlation 0.")
22 call disp%show(
"gramian = reshape([1., 0.5, 0.5, 1.], [2,2]) ! correlation 0.5")
23 gramian
= reshape([
1.,
0.5,
0.5,
1.], [
2,
2])
24 call disp%show(
"exp(getLogVolEll(gramian))")
29 call disp%show(
"gramian = reshape([1., 0.99, 0.99, 1.], [2,2]) ! correlation 0.99")
30 gramian
= reshape([
1.,
0.99,
0.99,
1.], [
2,
2])
31 call disp%show(
"exp(getLogVolEll(gramian))")
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind typeparameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in CFortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in CFortran Intero...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort fpp standardsemantics O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standardsemantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran cpp ffreelinelengthnone O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example output ⛓
2gramian
= reshape([
1.,
0.,
0.,
1.], [
2,
2])
7gramian
= reshape([
1.,
0.5,
0.5,
1.], [
2,
2])
12gramian
= reshape([
1.,
0.99,
0.99,
1.], [
2,
2])
 Test:
 test_pm_ellipsoid
 Todo:
 High Priority: A positivedefiniteness runtime check for the
gramian
input argument of this generic interface must be added.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.

If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
 Copyright
 Computational Data Science Lab
 Author:
 Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin
Definition at line 746 of file pm_ellipsoid.F90.